Functional Itô Calculus and Volatility Risk Management Bruno Dupire Bloomberg L.P/NYU AIMS Day 1 Cape Town, February 17, 2011

Outline Functional Itô Calculus Functional Itô formula Functional Feynman-Kac PDE for path dependent options 2) Volatility Hedge Local Volatility Model Volatility expansion Vega decomposition Robust hedge with Vanillas Examples

1) Functional Itô Calculus

Why?

Review of Itô Calculus 1D nD infiniteD Malliavin Calculus current value 1D nD infiniteD Malliavin Calculus Functional Itô Calculus possible evolutions

Functionals of running paths 12.87 6.34 6.32 T

Examples of Functionals

Derivatives

Examples

Topology and Continuity s Y X

Functional Itô Formula

Fragment of proof

Functional Feynman-Kac Formula

Delta Hedge/Clark-Ocone

P&L of a delta hedged Vanilla Break-even points Option Value Delta hedge

Functional PDE for Exotics

Classical PDE for Asian

Better Asian PDE

2) Robust Volatility Hedge

Local Volatility Model Simplest model to fit a full surface Forward volatilities that can be locked

Summary of LVM Simplest model that fits vanillas In Europe, second most used model (after Black-Scholes) in Equity Derivatives Local volatilities: fwd vols that can be locked by a vanilla PF Stoch vol model calibrated  If no jumps, deterministic implied vols => LVM

S&P500 implied and local vols

S&P 500 Fit Cumulative variance as a function of strike. One curve per maturity. Dotted line: Heston, Red line: Heston + residuals, bubbles: market RMS in bps BS: 305 Heston: 47 H+residuals: 7

Hedge within/outside LVM 1 Brownian driver => complete model Within the model, perfect replication by Delta hedge Hedge outside of (or against) the model: hedge against volatility perturbations Leads to a decomposition of Vega across strikes and maturities

Implied and Local Volatility Bumps implied to

P&L from Delta hedging

Model Impact

Comparing calibrated models

Volatility Expansion in LVM

Fréchet Derivative in LVM

One Touch Option - Price Black-Scholes model S0=100, H=110, σ=0.25, T=0.25

One Touch Option - Γ

Up-Out Call - Price Black-Scholes model S0=100, H=110, K=90, σ=0.25, T=0.25

Up-Out Call - Γ

Black-Scholes/LVM comparison

Vanilla hedging portfolio I

Vanilla hedging portfolios II

Example : Asian option K T K T

Asian Option Hedge K T K T

Forward Parabola - Γ Payoff: = 1, = 2, = 100, = 10 σ

Forward Parabola Hedge

Forward Cube - Γ Payoff: = 1, = 2, = 100, = 10 σ

Forward Cube Hedge

Forward Start - Γ Payoff = 1, = 2, = 100, = 10 σ

Forward Start Hedge

One Touch - Γ Payoff = 1, = 104 , = 100, = 5 σ

One Touch Hedge

Robustness Superbucket hedge protects today from first order moves in implied volatilities; what about robustness in the future? After delta hedge, the tracking error is In the case of discrete hedging in the model or fast mean reversion of vol, variance of TE is proportional to

Hedgeability The optimal vanilla hedge PF minimizes and thus coincides with the superbucket hedge as The residual risk is The hedgeability of an option is linked to the dependency of the functional gamma on the path

The Geometry of Hedging Bruno Dupire

In practice, determining the best hedge is not sufficient  Need to measure of residual risk

H-Squared Parabola vs Cube

Skewness Products Negative Skewness: fat tail to the left Linked to UP Dispersion < Down Dispersion Some products are based on Up Dispersion – Down Dispersion

Hedge with Components Decorrelated Case Good hedge with risk reversals on the components

Hedge with Components Correlated Case No hedge with components

BMW Sample Mean of Best third + Sample Mean of Worst third -2 x Sample Mean of Middle third

Perfect Hedge for BMW, Correlation = 0% Short 6 Shares, Short 9 Puts at –K*, Long 9 Calls at K*

BMW with 30 stocks , Correlation = 0 Hedge with Puts and Calls using 120 strikes R-squared = 80.7%, MC runs = 10,000

BMW with 30 stocks , Correlation = 40% Hedge with Puts and Calls using 120 skrikes R-squared = 5.8%, MC runs = 10,000

Analysis Correlation shifts the returns, which affects the components hedge but not the target In the absence of hedge, it is a mistake to price the product with a calibrated model Implied individual skewness is irrelevant ! It is a wrong way to play implied versus realized skewness

Conclusion Itô calculus can be extended to functionals of price paths Price difference between 2 models can be computed We get a variational calculus on volatility surfaces It leads to a strike/maturity decomposition of the volatility risk of the full portfolio